A triangle \(ABC\) is inscribed in a conic \(S\). The tangents to \(S\) at \(B\) and \(C\) meet in \(A'\), and \(B', C'\) are similarly defined. \(L\) is an arbitrary point of the line \(BC\); \(C'L\) meet in \(M\) and \(B'L\) meet in \(N\). Prove that \(M, N, A'\) are collinear and that the triangle \(LMN\) is self-polar with respect to \(S\).
The cartesian coordinates of the points of a hyperbola are expressed in the parametric form \((p\theta+q\theta^{-1}+r, p'\theta+q'\theta^{-1}+r')\), where \(p,q,r,p',q',r'\) are fixed and \(pq' \neq p'q\). Find (i) the equation of the tangent to the hyperbola at a general point \(\theta\), (ii) the equations of the asymptotes, (iii) the coordinates of the centre.
Find the necessary and sufficient condition that the two pairs of lines \[ ax^2+2hxy+by^2=0, \quad a'x^2+2h'xy+b'y^2=0 \] should be harmonically conjugate. Prove that, if \(\lambda\mu\nu=1\), any tangent to one of the conics \[ x^2+2\lambda y=0, \quad y^2+2\mu x=0, \quad 2\nu xy+1=0, \] cuts the other two conics in harmonically conjugate pairs of points.
A series of circles is drawn with given centre \(O\). Show that the mid-points of their chords of intersection with a given central conic \(S\) lie on a rectangular hyperbola \(\Gamma\), whose asymptotes are parallel to the axis of \(S\) and which passes through \(O\) and the centre of \(S\). Show further that the same rectangular hyperbola \(\Gamma\) is obtained if \(S\) is replaced by any conic \(S'\) which passes through the common points of \(S\) and any circle with centre \(O\).
The lines joining a point \(O\) in the plane of a triangle \(ABC\) to the vertices meet the sides \(BC, CA,\) and \(AB\) in \(D, E,\) and \(F\) respectively. If the circumcircle of the triangle \(DEF\) meets \(BC, CA,\) and \(AB\) again in \(P, Q,\) and \(R\) respectively, prove that \(AP, BQ,\) and \(CR\) are concurrent. Prove also that if the circumcircle of \(DEF\) meets \(OA\) again in \(U\), and \(OB\) again in \(V\), then \(AV\) and \(BU\) meet on \(OR\).
Prove for any tetrahedron that the perpendicular from a vertex on to the opposite face will meet the three lines drawn perpendicular to and through the orthocentres of the other three faces. Prove that if two perpendiculars from vertices on to opposite faces meet in a point \(O\), then the two perpendiculars through the orthocentres of the other two faces will also meet in \(O\). Prove that if three perpendiculars from vertices meet in \(O\), the fourth will also pass through \(O\), and the four perpendiculars at the orthocentres will also meet in \(O\). State the relationship in this last case between pairs of opposite edges of the tetrahedron.
Define inversion in plane geometry and show that orthogonal curves are inverted into orthogonal curves. The points \(P, P'\) are inverse points with respect to the circle \(C\), and the whole figure is inverted with respect to a point \(O\). Prove that \(P\) and \(P'\) are inverted into two points which are inverse with respect to the circle into which \(C\) is inverted. What happens if \(O\) lies on the circumference of \(C\)? Hence, or otherwise, prove that the centres of the circumcircles of the four triangles formed by four given straight lines are concyclic.
Prove that the locus of a point moving so that the lengths of tangents drawn from it to two fixed circles \(C_1\) and \(C_2\) are in constant ratio is a third circle \(C_3\) coaxal with \(C_1\) and \(C_2\). If for three coaxal circles \(\lambda_{12}\) denotes the fixed ratio of lengths of tangents from a point of \(C_3\) to \(C_1\) and \(C_2\) respectively, and if \(\lambda_{23}\) and \(\lambda_{31}\) are similarly defined, prove that \(\lambda_{12}.\lambda_{23}.\lambda_{31}=1\).
Find the condition for rectangular cartesian tangential coordinates that the line equation of the second degree: \[ Al^2 + Bm^2 + Cn^2 + 2Fmn + 2Gnl + 2Hlm = 0 \] should represent a parabola. Prove that the locus of points from which perpendicular tangents can be drawn to the general conic is the circle whose point equation is \[ C(x^2+y^2) - 2Gx - 2Fy + A+B=0, \] and identify the locus in the case of the parabola. Show that the circles obtained from conics touching four fixed straight lines form a coaxal set, and identify the radical axis of the set.
Prove that the mid-points of parallel chords of a conic lie on a straight line. Show that the locus of feet of perpendiculars on to parallel chords of a conic from their respective poles is, in general, a rectangular hyperbola. Identify the asymptotes of the rectangular hyperbola and show that the curve meets the original conic in points where the normal is either parallel to or perpendicular to the direction of the chords.